Semiabelian group

Not to be confused with semiabelian scheme.

Semiabelian groups are a class of groups first introduced by Thompson (1984) and named by Matzat (1987).^[1] It appears in Galois theory, in the study of the inverse Galois problem or the embedding problem which is a generalization of the former.

Definition

Definition:^[2][3][4][5] A finite group G is called semiabelian if and only if there exists a sequence

${\displaystyle G_{0}=\{1\},G_{1},\dots ,G_{n}=G}$

such that ${\displaystyle G_{i}}$ is a homomorphic image of a semidirect product ${\displaystyle A_{i}\rtimes G_{i-1}}$ with a finite abelian group ${\displaystyle A_{i}}$ (${\displaystyle i=1,\dots ,n}$.).

The family ${\displaystyle {\mathcal {S}}}$ of finite semiabelian groups is the minimal family which contains the trivial group and is closed under the following operations:^[6][7]

• If ${\displaystyle G\in {\mathcal {S}}}$ acts on a finite abelian group ${\displaystyle A}$, then ${\displaystyle A\rtimes G\in {\mathcal {S}}}$;
• If ${\displaystyle G\in {\mathcal {S}}}$ and ${\displaystyle N\triangleleft G}$ is a normal subgroup, then ${\displaystyle G/N\in {\mathcal {S}}}$.
  

The class of finite groups G with a regular realizations over ${\displaystyle \mathbb {Q} }$ is closed under taking semidirect products with abelian kernels, and it is also closed under quotients. The class ${\displaystyle {\mathcal {S}}}$ is the smallest class of finite groups that have both of these closure properties as mentioned above.^[8][9]

Example

• Abelian groups, dihedral groups, and all p-groups of order less than ${\displaystyle 64}$ are semiabelian.^[10]
• The following are equivalent for a non-trivial finite group G (Dentzer 1995, Theorem 2.3.):^[11][12]

  (i) G is semiabelian.

  (ii) G possess an abelian ${\displaystyle A\triangleleft G}$ and a some proper semiabelian subgroup U with ${\displaystyle G=AU}$.

Therefore G is an epimorphism of a split group extension with abelian kernel.^[13]

• Finite semiabelian groups possess G-realizations^[14][15] over function fields ${\displaystyle k(t)}$ in one variable for any field ${\displaystyle k}$ and therefore are Galois groups over every Hilbertian field.^[16]